TY  - THES
AB  - This dissertation develops a functional-analytic framework for discrete-time dynamical systems by studying Koopman and Perron-Frobenius operators in both topological and measure-preserving settings.It establishes new operator-theoretic characterizations of geometric properties of the underlying dynamics, including injectivity, density of image, surjectivity, and homeomorphism, and connects these to ergodic decompositions and mixing behavior.A central contribution is the analysis of entropy through operator approximation: the work relates measure-theoretic entropy, Voiculescu’s approximation entropy, and newly introduced fractional approximation bounds, showing in particular that positive measure-theoretic entropy enforces unbounded approximation complexity.From a data-driven perspective, the thesis studies extended Dynamic Mode Decomposition and proves quantitative lower bounds on the dimension required for accurate finite-time Koopman prediction, with linear and exponential growth laws governed by approximation and measure entropy.Finally, it introduces Banach lattice entropy as a unifying notion recovering both Kolmogorov-Sinai and topological entropy, and characterizes uniform ergodicity of Koopman operators in terms of eventual periodicity (topological case) and periodicity (measure-preserving case).
AU  - Hölz, Julian
CY  - Wuppertal
DA  - 2026
DO  - 10.25926/BUW/0-1013
DP  - Bergische Universität Wuppertal
KW  - Functional-Analysis
KW  - Ergodic Theory
KW  - Dynamical Systems
KW  - Dynamic Mode Decomposition
KW  - Koopman Operator
KW  - Uniform Ergodicity
KW  - Entropy
KW  - Measure-Theoretic Entropy
KW  - Topological Entropy
KW  - Approximation Entropy
KW  - Voiculescu-Brown Entropy
KW  - Fractional Approximation Entropy
LA  - eng
N1  - Bergische Universität Wuppertal, Dissertation, 2026
PB  - Veröffentlichungen der Universität
PY  - April 2026
SP  - 1 Online-Ressource (iv, 191 Seiten)
T2  - Fakultät für Mathematik und Naturwissenschaften
TI  - Characterization, long-term behavior, and approximation of Koopman operators
UR  - https://nbn-resolving.org/urn:nbn:de:hbz:468-2-6991
Y2  - 2026-05-16T11:37:39
ER  -